Abstract
The \({{\mathrm{\mathrm {AT}}}}\)-free order is a linear order of the vertices of a graph the existence of which characterizes \({{\mathrm{\mathrm {AT}}}}\)-free graphs. We show that all \({{\mathrm{\mathrm {AT}}}}\)-free orders of an \({{\mathrm{\mathrm {AT}}}}\)-free graph can be generated in O(1) amortized time.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
T. Kloks—This author thanks the institute for their hospitality and support.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A convexity space has Carathéodory number k if k is the smallest integer satisfying the following property. If \(x \in \sigma (X)\) for a set X then there exists a \(X^{\prime } \subseteq X\) with \(|X^{\prime }| \le k\) and \(x \in \sigma (X^{\prime })\).
References
Björner, A., Ziegler, G.: 8 Introduction to greeds. In: White, N. (ed.) Matroid Applications - Encyclopedia of Mathematics and its Applications, vol. 40, pp. 284–357. Cambridge University Press, Cambridge (1992)
Calder, J.: Some elementary properties of interval convexities. J. Lond. Math. Soc. 3, 422–428 (1971)
Chandran, L., Ibarra, L., Ruskey, F., Sawada, J.: Generating and characterizing the perfect elimination orderings of a chordal graph. Theor. Comput. Sci. 307, 303–317 (2003)
Chang, J.-M., Ho, C.-W., Ko, M.-T.: LexBFS-ordering in asteroidal triple-free graphs. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 163–172. Springer, Heidelberg (1999)
Chvátal, V.: Antimatroids, betweenness, convexity. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization. Bonn, 2009, pp. 57–64. Springer, Heidelberg (2008)
Corneil, D., Stacho, J.: Vertex ordering characterizations of graphs of bounded asteroidal number. J. Graph Theor. 78, 61–79 (2015)
Kloks, T., Kratsch, D., Müller, H.: Asteroidal sets in graphs. In: Möhring, Rolf H. (ed.) WG 1997. LNCS, vol. 1335, pp. 229–241. Springer, Heidelberg (1997)
Kloks, T., Wang, Y.: Advances in graph algorithms. Manuscript viXra:1409.0165 (2014)
Pruesse, G., Ruskey, F.: Gray codes from antimatroids. Order 10(3), 239–252 (1993)
Pruesse, G., Ruskey, F.: Generating linear extensions fast. SIAM J. Comput. 23, 373–386 (1994)
Sawada, J.: Oracles for vertex elimination orderings. Theor. Comput. Sci. 341, 73–90 (2005)
Walter, J.: Representations of chordal graphs as subtrees of a tree. J. Graph Theor. 2, 265–267 (1978)
Acknowledgments
The authors would like to thank the anonymous reviewers for the comments. Jou-Ming Chang was supported in part by the MOST grant 104-2221-E-114-002-MY3. Hung-Lung Wang was supported in part by the MOST grant 104-2221-E-114-003.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Chang, JM., Kloks, T., Wang, HL. (2016). Gray Codes for AT-Free Orders via Antimatroids. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-29516-9_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29515-2
Online ISBN: 978-3-319-29516-9
eBook Packages: Computer ScienceComputer Science (R0)